Electron-electron interaction correction to the skew scattering contribution to the anomalous Hall effect in ultrathin FeCo films

Electron-electron interaction (EEI), a quantum effect that occurs in low-dimensional or disordered systems at low temperatures, is an important issue in electron transport phenomena, especially in the anomalous Hall effect (AHE). Nevertheless, the role of EEI correction to the AHE is overlooked in conventional magnetic metal systems and the scaling exponents of the power-law fits are not consistently at the systems with sufficiently high disorder. Here, we demonstrate that EEI does correct to the AHE and cause $\text{ln}T$-type temperature dependence of sheet resistance $R_{xx}$ (conductance $G_{xx}$), anomalous Hall resistance $R_{\mathrm{AH}}$, and anomalous Hall conductance $G_{\mathrm{AH}}$ in ultrathin high-quality FeCo films in the low-temperature region. Furthermore, the scaling exponent γ ∼ 1 is acquired through the power-law fits of $G_{\mathrm{AH}}\sim G_{xx}^{\gamma }$ by varying temperature or thickness, suggesting that EEI corrects to skew scattering and definitely results in skew scattering dominating AHE in the bad metal regime (${\sigma }_{xx}<1\times {10}^4$ S/cm).

Figure 1

Surface morphology of FeCo films. RHEED patterns of the MgO(001) and FeCo surfaces taken along azimuths (a) MgO[100], (b) MgO[110], (c) 1.0-nm FeCo[110], (d) 1.0-nm FeCo[100], (e) 3.0-nm FeCo[110], and (f) 3.0-nm FeCo[100], respectively.

Figure 2

Crystal quality and surface scattering of FeCo films. (a) Symmetric and (b) asymmetric HRXRD spectra for 1.0–3.0-nm FeCo films. (c) HRXRD azimuthal ϕ scans of the bcc FeCo(101) and MgO(202) planes for 3.0-nm FeCo films. (d) Thickness $d$ dependence of the residual resistivity ${\rho }_{xx0}$ for 1.0–3.0-nm FeCo films. Inset: The ${\rho }_{xx0}$ as a function of $d$ for 1.0–43.8-nm FeCo films. Note that the data of 4.1–43.8-nm FeCo films are obtained from our previous work [36]. The red lines are a guide for the eye.

Figure 3

Quantum corrections to sheet resistance and AH resistance. [(a), (b)] Temperature dependence of the normalized sheet resistance $R_{xx}$ and anomalous Hall resistance $R_{\mathrm{AH}}$ for FeCo films with various thicknesses, respectively. (c) $G_{xx}$ as a function of ln($T/T_0$) for 1.6-nm FeCo films. $T_0$ is taken as 60 K, which corresponds to the conductance maximum. The red line is the linear fit according to Eq. (1), and the dependence of $d{G}_{xx}/d\ln (T/T_0)$ on thickness $d$ of FeCo films is shown in inset. (d) $G_{xx}$ as a function of $\text{ln}T$ for 1.6-nm FeCo films in magnetic fields (0, 3, 6, and 9 T). The inset shows $d{G}_{xx}/d\ln T$ as a function of magnetic field for FeCo films with different thickness.

Figure 4

EEI correction to AH conductance. (a) $\text{ln}T$ dependence of normalized relative changes in the sheet resistance $R_{xx}$, anomalous Hall resistance $R_{\mathrm{AH}}$, and anomalous Hall conductance $G_{\mathrm{AH}}$ for 1.6-nm FeCo films. The straight lines are the linear fits to the data from 2 to 20 K. (b) Prefactors $A_{\mathrm{R}}, A_{\mathrm{AH}}, A_{\mathrm{G}}$, and $2A_{\mathrm{R}}-A_{\mathrm{AH}}$ as a function of thickness $d$ for FeCo films.

Figure 5

Scaling of AHE by varying temperature. [(a)–(c)] Log-log plot of the anomalous Hall resistance $R_{\mathrm{AH}}$ of 1.0-, 1.3-, and 1.6-nm FeCo films as a function of the sheet resistance $R_{xx}$ with varying temperature. $R_{\mathrm{Q}}=h/e^2$ is the quantum resistance. The solid lines provide a guide for the eye. [(d)–(f)] Log-log plot of the anomalous Hall conductance $G_{\mathrm{AH}}$ of 1.0-, 1.3-, and 1.6-nm FeCo films as a function of the sheet conductance $G_{xx}$ with varying temperature. The solid lines are the separate power-law fits to the data from 2 to 20 K.

Figure 6

Scaling of AHE by varying sample thickness. (a) Log-log plot of the anomalous Hall resistance $R_{\mathrm{AH}}$ of FeCo films as a function of the sheet resistance $R_{xx}$ at 2 K. The solid lines provide a guide for the eye. (b) Log-log plot of the anomalous Hall conductance $G_{\mathrm{AH}}$ as a function of the sheet conductance $G_{xx}$ at 2 K. The solid lines are separate power-law fits to the low- and high-conductance data, giving exponents of $\gamma =1.01\pm 0.05$ and $\gamma =0.74\pm 0.01$, respectively. Note that the data of 4.1–43.8-nm FeCo films are obtained from our previous work [36].